The Sherrington-Kirkpatrick (SK) model is a mean field spin glass invented initially with the aim of explaining the strange magnetic behaviors of certain alloys. In 1979, G. Parisi predicted a variational formula for the thermodynamic limit of the free energy in the SK model and explained the role played by its minimizer, the Parisi measure. In this talk, we will discuss some of its conjectured properties and present partial results along this direction. In particular, we will establish the uniqueness of this measure by proving the strict convexity of the Parisi functional. This talk is based on a joint work with A. Auffinger.